 On the Fourier structure of the zero set of fractional Brownian motionWillem L. Fouché and Safari Mukeru Statistics & Probability Letters, 2013, vol. 83, issue 2, 459-466 Abstract: In this paper, we consider a one-dimensional fractional Brownian motion X and the Fourier transform of its associated Dirac measure δ(X). It is a measure, canonically associated with X (in the sense of Schwartz’s theory of generalized functions). As was shown by Kahane, this measure reflects the fractal geometry of the zero set of X to a remarkable degree. One can also think of this measure as being given by the local time of X at 0. Kahane initiated the problem of understanding the Fourier structure of the zero sets of fractional Brownian motion. In this paper, we address this problem by analysing the even moments of the Fourier transform of the Dirac measure of a one-dimensional fractional Brownian motion restricted to compact intervals. We shall represent these moments by Fresnel-type oscillatory integrals. In the case of Brownian motion, the second-order moment reveals close connections with Bessel functions and Fresnel integrals. Keywords: Fractional Brownian motion; Local times; Hausdorff dimension; Salem sets (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:eee:stapro:v:83:y:2013:i:2:p:459-466 Ordering information: This journal article can be ordered from DOI: 10.1016/j.spl.2012.10.015 Access Statistics for this article Statistics & Probability Letters is currently edited by Somnath Datta and Hira L. Koul More articles in Statistics & Probability Letters from Elsevier |
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